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For all functions $f:\mathbb{R}\to\mathbb{R}$ one can find a unique decomposition $f(x)=E(x)+O(x)$ where $E(-x)=E(x)$ and $O(-x)=-O(x)$.

Is there any branch of mathematics where analysing the decomposition of a function into its odd and even parts has an important role?

The even-odd decomposition of $e^x$ is $\cosh(x)+\sinh(x)$, and $e^{ix}$ has a decomposition into even and odd parts $\cos(x)+i\sin(x)$. Are there other well-known functions whose decompositions are other well-known functions?

My first question has been answered already here in detail, but it doesn't provide any answers for the second half of my question, that is examples of well-known functions with an interesting decomposition.

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I think the most important role in decomposing into even and odd parts is to make simplifications and reduce calculations.

Integrals:

For example, if you have $f(x)$ a odd function and you have to integrate

$$I = \int_{-a}^{a} f(x) \ dx$$

You know it's equal to zero at once, cause $f$ is odd:

$$I = \int_{-a}^{0} f(x) \ dx + \int_{0}^{a} f(x) \ dx = -\int_{0}^{a}f(y) \ dy + \int_{0}^{a} f(x) \ dx = 0$$

You can do the same for even functions: Instead of computing the integral over two intervals, you can compute twice the same interval:

$$\int_{-a}^{a} g(x) \ dx = 2 \cdot \int_{0}^{a} g(x) \ dx$$

Finding roots, minimum and maximum:

When are dealing with an even function $g(x)$, and you want to find the roots of $g$ at the interval $\left[a, \ b\right]$ with $a < 0 < b$.

Instead of searching in all domain, you can search in $\left[0, \ \max(-a, \ b)\right]$. Once you find a root $r$, then $(-r)$ will also be.

The same happens if you are searching for the minimum or the maximum of a function, which are in fact a problem of finding roots.

Computational graphics:

In computational graphics, if you know there's symmetry in relation to some axes (which is associated with a even function), instead of calculating the color every pixel in both sides, you can only copy and paste, reducing the computational cost.

Linear algebra:

The same idea appears not only in functions. For example, if you have any matrix $A$, you can decompose into a symmetric matrix (related to even) $D$ and a anti-symmetric matrix (related to odd) $W$. $$A = D + W$$ $$D = \dfrac{1}{2}\left(A + A^{T}\right)$$ $$W = \dfrac{1}{2}\left(A - A^{T}\right)$$

There are some applications like shown here

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